Magento on DigitalOcean server

Question

I have a problem with Magento store running on Digital Ocean server.

This is my droplet configuration:

2GB Ram | 40GB SSD Disk | New York 2 | Ubuntu Ubuntu 12.04.3 x64

When I am using Magento store, it works normally, except when I try to create account or do a purchase.

Then it takes 2 minutes to complete purchase.

I downloaded the whole site locally (with db) and the order and create account takes only few seconds.

I tried with upgrading droplet to 4GB Ram, but still the same.

This is my php configuration:

max_execution_time  30
max_file_uploads    20  20
max_input_nesting_level 64
max_input_time  60
max_input_vars  1000    
memory_limit    512M

I am not sure how to proceed with debugging this. Can someone advice?

Update #2 (based on UPD from MageWorx): Based on MageWorx(updated) suggestion, I ran mysqltunner on the server. These are the results:

[OK] Logged in using credentials from debian maintenance account.
[OK] Currently running supported MySQL version 5.5.38-0ubuntu0.12.04.1
[OK] Operating on 64-bit architecture

-------- Storage Engine Statistics -------------------------------------------
[--] Status: +ARCHIVE +BLACKHOLE +CSV -FEDERATED +InnoDB +MRG_MYISAM 
[--] Data in MyISAM tables: 25K (Tables: 28)
[--] Data in InnoDB tables: 36M (Tables: 1386)
[--] Data in PERFORMANCE_SCHEMA tables: 0B (Tables: 17)
[--] Data in MEMORY tables: 0B (Tables: 68)
[!!] Total fragmented tables: 1387

-------- Security Recommendations  -------------------------------------------
[OK] All database users have passwords assigned

-------- Performance Metrics -------------------------------------------------
[--] Up for: 49s (338 q [6.898 qps], 71 conn, TX: 181K, RX: 51K)
[--] Reads / Writes: 92% / 8%
[--] Total buffers: 832.0M global + 2.7M per thread (100 max threads)
[OK] Maximum possible memory usage: 1.1G (54% of installed RAM)
[OK] Slow queries: 0% (0/338)
[OK] Highest usage of available connections: 2% (2/100)
[OK] Key buffer size / total MyISAM indexes: 32.0M/178.0K
[!!] Query cache efficiency: 12.9% (30 cached / 233 selects)
[OK] Query cache prunes per day: 0
[OK] Sorts requiring temporary tables: 0% (0 temp sorts / 12 sorts)
[OK] Temporary tables created on disk: 25% (57 on disk / 222 total)
[OK] Thread cache hit rate: 97% (2 created / 71 connections)
[OK] Table cache hit rate: 25% (1K open / 6K opened)
[OK] Open file limit used: 1% (104/8K)
[OK] Table locks acquired immediately: 100% (248 immediate / 248 locks)
[OK] InnoDB buffer pool / data size: 512.0M/36.8M
[OK] InnoDB log waits: 0
-------- Recommendations -----------------------------------------------------
General recommendations:
    Run OPTIMIZE TABLE to defragment tables for better performance
    MySQL started within last 24 hours - recommendations may be inaccurate
    Enable the slow query log to troubleshoot bad queries
Variables to adjust:
    query_cache_limit (> 4M, or use smaller result sets)

These are the results from apachebuddy:

Your server has 2002 MB of memory
The largest apache process is using 39.78 MB of memory
The smallest apache process is using 15.04 MB of memory
The average apache process is using 17.32 MB of memory
Going by the average Apache process, Apache can potentially use 433.00 MB RAM (21.63 % of available RAM)
Going by the largest Apache process, Apache can potentially use 994.50 MB RAM (49.68 % of available RAM)

Generating reports...
### GENERAL REPORT ###

Settings considered for this report:

    Your server's physical RAM:     2002MB
    Apache's MaxClients directive:      25
    Apache MPM Model:           prefork
    Largest Apache process (by memory): 39.78MB
[ OK ]  Your MaxClients setting is within an acceptable range.
    Max potential memory usage:         994.5 MB

    Percentage of RAM allocated to Apache   49.68 %

Answer 1

UPD:

As I can see from the results you got with mysqltuner, you need to experiment with the following settings:

skip-networking
query_cache_limit = 4M
query_cache_size = 256M

thread_concurrency = 4
table_open_cache = 4096
innodb_buffer_pool_size = 512M
join_buffer_size = 1M

Also note that after changes are made to the my.cnf file, don't forget to restart mysql

sudo service mysql restart

After settings are applies, test your website for some time, and then launch mysqltuner to see the results.