In EVE Online:
My favorite all-around ship is the Hurricane. It can handle a wide variety of situations in both PVE and PVP. It doesn't have the defensive power of the Drake or a battleship, but it makes up for it with being very fast and having about 1.75x the damage output of the Drake. It was nerfed recently (with good reason, as it was a little OP) to make it a little fairer when pitched against the other ships in its class, but the overall playstyle that the Hurricane follows is still my favorite. Its upgraded tier, the Republic Fleet Hurricane (which is the old un-nerfed Hurricane) is basically the regular 'cane, but better, and it can go toe-to-toe with nearly any sub-capital and come out victorious. Its weakness is stasis webifiers, which take away its dodging ability ("Speed tank"), but that can be countered with fleet support or having some onboard EWAR or capacitor drainers.
For PVP, I generally like the versatile Rifter. It can be fitted for speed, offense, or defense. Obviously it's outclassed by higher-tier ships like the Dramiel or its T2 variants or Interceptors, but the Rifter was my first PVP ship and will always be one of my favorites. Also, low cost can be a significant advantage because it allows you to take some pretty stupid risks with it, and that always results in a good time.